On the linkage between the k−5/3 spectral and k−7/3 cospectral scaling in high-Reynolds number turbulent boundary layers

Abstract

Connections between the “−5/3” spectral and “−7/3” cospectral scaling exponents characterizing the inertial subranges of the wall-normal energy spectrum and the turbulent momentum flux cospectrum are explored in the equilibrium layer of high-Reynolds number turbulent boundary layers. Previous laboratory experiments and field measurements featured here in the atmospheric boundary layer show that the “−7/3” scaling in the momentum flux cospectrum Fuw(k) commences at lower wavenumbers (around kz = 3) than the “−5/3” scaling in the wall-normal energy spectrum Eww(k) (around kz = 6), where k is the streamwise wavenumber and z is the distance from the surface. A satisfactory explanation as to why Fuw(k) attains its “−7/3” inertial subrange scaling earlier than Eww(k) in wavenumber space remains elusive. A cospectral budget (CSB) model subject to several simplifications and closure schemes offers one viewpoint. In its simplest form, the CSB model assumes a balance at all k between the production term and a Rotta-like pressure decorrelation term with a prescribed wavenumber-dependent relaxation time scale. It predicts the “−7/3” scaling for Fuw(k) from the “−5/3” scaling in Eww(k), thereby recovering earlier results derived from dimensional considerations. A finite flux transfer term was previously proposed to explain anomalous deviations from the “−7/3” cospectral scaling in the inertial subrange using a simplified spectral diffusion closure. However, this explanation is not compatible with an earlier commencement of the “−7/3” scaling in Fuw(k). An alternative explanation that does not require a finite flux transfer is explored here. By linking the relaxation time scale in the slow-component of the Rotta model to the turbulent kinetic energy (TKE) spectrum, the earlier onset of the “−7/3” scaling in Fuw(k) is recovered without attainment of a “−5/3” scaling in Eww(k). The early onset of the “−7/3” scaling at smaller k is related to a slower than k−5/3 decay in the TKE spectrum at the crossover from production to inertial scales.